Product structures for Legendrian contact homology

Abstract

Abstract. Legendrian contact homology (LCH) and its associated differential graded algebra are powerful non-classical invariants of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and noncommutative) information. To recover some of the nonlinear information while preserving computability, we introduce invariant cup and Massey products -and, more generally, an A∞ structure -on the linearized LCH. We apply the products and A∞ structure in three ways: to find infinite families of Legendrian knots that are not isotopic to their Legendrian mirrors, to reinterpret the duality theorem of the fourth author in terms of the cup product, and to recover higher-order linearizations of the LCH.

Bilinearized Legendrian contact homology and the augmentation category

“…Those are constructed as a generalisation of linearised Legendrian contact homology using two augmentations instead of one. Considering similar constructions with more augmentations leads to the higher order compositions map in the category and generalises the idea of [6] where an A ∞ -algebra was constructed from one augmentation. This category allows us to define a notion of equivalence of augmentations when the coefficient ring is a field regardless of its characteristic.…”

Cellular Legendrian contact homology for surfaces, part II

“…The set of isomorphism types of the rings {LCH * (L, ǫ) | ǫ any graded augmentation of A} depends only on the stable tame isomorphism type of (A, ∂) and is hence a Legendrian isotopy invariant [6].…”

Massey products, A∞-algebras, differential equations, and Chekanov homology

“…Classical (in the sense of [19]) Massey products on linearized Chekanov homology have been used earlier in [5] to distinguish between some Legendrian knots and their mirror images. The observation that matric and generalized Massey products, as in [2,21], also yield Legendrian invariants seems to be new.…”

“…The naturality of classical and generalized Massey products (Propositions 3.3 and 3.5) yields the following result: Massey product invariants are useful. Civan and his coworkers prove in [5] the existence of an A ∞ -algebra structure on a linearized complex LC(K) built from the Chekanov algebra, see [4], and they show that there exists an infinite family of knots that are distinguishable from their Legendrian mirrors by using classical Massey products on the cohomology of LC(K).…”